- The number of party guests increases according to the function g(t) = 2t + 4, where t is the number of hours after the party started and g is the number of guests.
- The number of iPads sold increases according to the function s(t) = 2t + 4, where t is the number of weeks after the iPad went on sale and s is the number of iPads sold in millions.
- The number of points the team scores increases according to the function p(t) = 2t + 4, where t is the number of minutes after halftime and p is the number of points scored.

Party guests. iPads. Points.

They’re all the same to the student who doesn’t understand abstraction, the process by which we turn those contexts into words and symbols. The idea that any one of those contexts will engage that student any more than another is a fiction.

**BTW**. Clarifying: the issue at hand isn’t that these three problems are simplistic or false abstractions of a context. It’s that they *start* at a high-level of abstraction. (This isn’t a revisitation of pseudocontext, in other words.)

**2012 Sep 27**. Nathan Kraft points us to some research that says, “This kind of superficial personalization, indeed, increases engagement and achievement.” So I may have overstated my case considerably. The point of this ladder of abstraction series, though, is that investments in making abstraction more explicit are way more worth our while, not that other investments aren’t important also.

## 13 Comments

## Nico

September 24, 2012 - 8:03 am -Context:

My brother and I used to love playing street hockey*. A tennis ball was usually the ‘puck’ of choice. We had one. It was old, didn’t have much tennis bounce and the rain/dirt had turned it closer to brown than the original fluorescent tennis ball yellow. (but still, it counts as 1)

One Christmas my mom gave me the gift of new tennis balls.

They come in a cylindrical package of 3 (this is often common knowledge).

What if my mom had given me 2 packages? (one for each brother) Or if she really really loved me (if one measures love by gifts) and she gave me 42 packages of tennis balls?

Can I get my students to**:

T(p) = 3p +1

T= # of tennis balls

p= # of packages given as gifts

* This is a true story

** Does the truth of it make this particular rung lower?

## May Trix (@DeeTerminant)

September 24, 2012 - 8:17 am -What about when a particular context (usually money in my experience) does engage a student and the teacher becomes convinced that the student understands how to generalize what they are doing to any context?

## Bob Lochel

September 24, 2012 - 8:54 am -It’s not only fiction to have students buy into these simple formulas, it’s flat-out deceptive, and conditions students to believe that a short, snappy function will model the behavior of data at all time. And of course the model will be linear for value of x…..of course it will!

We are often our own worst enemies for allowing these unrealistic problems to creep into our courses. I am currently working with teachers to adapt courses for the Pennsylvania Keystone exams in Algebra 1. Many of the released items feature functions like the ones you cite, so the natural response of teachers is to mimic them, under the mistaken impression that we are helping students learn algebra better. So worksheets are created, and we begin to feel good about ourselves that students can now interpret a rate of change when given a function. But if given a data set, or a method of collecting data, could our students now develop and adapt a model? Doubtful.

I find myself often fighting the efficiency battle with my colleagues: that organizing a data-collection experiment, buidling, adapting and communicating a model will be more instructive than having 50 worksheet problems. Some colleagues will bemoan the amount of time data collection will take, while others cling to the notion that worksheet math is actually helpful. It’s the latter group which scares me the most.

## Dan Meyer

September 24, 2012 - 9:08 am -Nico:The level of abstraction is independent of the truth value of the task. If the task starts with t(p) = 3p + 1, we are assuming that students are already comfortable with the abstraction of function notation. If they

aren’tcomfortable, it won’t matter if the task is true, false, tennis balls, or cupcakes.## Nico

September 24, 2012 - 11:51 am -Dan

-‘The level of abstraction is independent of the truth value of the task.’

I’m seeing this as true. But now can we include the level of engagement. One question would be, is the truth (or level personal attachment) in this (or that) story more engaging? The next question would be, is engaging the student getting us closer to a new rung on the ladder?

My main question though would be

I want my students to get to this rung:

>>t(p) = 3p + 1,

If my task starts with a story that eventually gets my students to create and understand this rung:

>The number of tennis balls will be equal to ONE plus THREE times the number of gifts I get from my mom.

Is this helping me as a teacher?

Am I closer to the next rung?

My initial feeling is that I am getting them closer to that higher rung. Am I lying to myself here? I am legitimately questioning this. I’m trying to build to 3p +1. Not start there.

## William

September 24, 2012 - 4:19 pm -Nico,

What if, instead of writing a story [1], you took a picture of a couple of loose tennis balls, and a stack of tennis ball cans. Ask your students to estimate how many tennis balls there are in the photo.

How do they count the number of balls in the cans? What information do they need to know about the cans? What do they need to know about the loose balls?

If you gave them another picture with some cans removed (or added), could they write a sentence to describe how many balls are in the scene? (You might want to have five or six pictures.)

I think something like that would guide your students towards the abstraction you’re looking for.

[1] It’s very hard to write good story problems, doubly so using words. The story has to be good as a story: there has to be motivation, tension, and resolution. Beyond that, the story has to embed in itself information that will cause students to construct some sort of mathematical abstraction that will be useful. It can be done, but it’s really hard.

The two difficulties with the story you suggested are that there’s no drama in it – why does it matter how many tennis balls you got? If you have one you can play as well as if you have seven or one hundred and twenty-seven. There’s nothing to make me wonder about the answer, nothing to perplex me.

Beyond that, a function is a counter-intuitive abstraction to use to answer it. Why wouldn’t I skip count by threes? Or just open the cans and actually count by ones? Functions are useful when something varies with respect to something else, but it doesn’t seem like the number of balls you’d get naturally varies.

## Nico

September 24, 2012 - 5:43 pm -William,

Thanks for the response.

On two difficulties about the story.

1) no drama.

For the sake of the discussion, I restrained the writing of the full story here. Nor did I tell it dramatically. I’ll leave that for the classroom. But your point is well met. Story telling with drama can be difficult.

2) function abstraction.

By asking, ‘what if I received 100 packages?’ the students might come up with 301. I might then ask how did they get that? And they would answer:

>The number of tennis balls will be equal to ONE plus THREE times the number of gifts.

So, is this statement (the achievement of this rung) a victory for me as a teacher? Am I closer to the goal of:

>>t(p) = 3p + 1 (the next rung)

On writing stories in general.

‘Itâ€™s very hard to write good story problems, doubly so using words.’

Yes. I agree here.

However, what if I claim the picture you describe seems less dramatic, with even less perplexity. Are we now debating whether it’s easier to write (find) a good story, versus take (find) a good picture?

Back to the original idea from Dan’s post, because I feel I’ve drifted away a bit.

Is the starting point of this story telling low enough on the ladder?

For further discussion we can move to mr_row at hotmail if you’d like.

In my case, starting with Dan’s example #1)

“The number of party guests increases according to the function g(t) = 2t + 4, where t is the number of hours after the party started and g is the number of guests”

would absolutely be too high for my grade 8 students.

But somewhere, not sure where, that level of abstraction would be common place. Say grade 10. Is that wrong?

Which rung should we start at could depend on who we’re asking?

r(g)? Is rung a function of grade level?

love this discussion.

cheers

## William

September 24, 2012 - 6:10 pm -Nico,

I’ll take a stab at answering your questions the way I think Dan would.

In the story you’ve told, you give your students exactly the right information they need to answer your question. Much of the value of working up and down the ladder of abstraction is learning to identify which information is useful for answering a particular question.

What if you didn’t tell them how many balls are in a can? Is that relevant? What if you didn’t tell them how many balls you had initially? Is that relevant? Having them think through those questions will guide them in constructing an abstraction to answer your question. (And I’m still not sure a function is a good abstraction to answer that; varying the number of gifts seems contrived, especially when it hits the 100 gift mark.)

“The number of party guests increases according to the function g(t) = 2t + 4, where t is the number of hours after the party started and g is the number of guests.”

The world doesn’t come in pre-digested chunks like this outside the math classroom. In reality, what happens is we walk into a party, and we see some guests already there and some guests filtering in. Maybe some filtering out. Maybe we take a picture every couple of minutes, maybe we record a video. That’s a (relatively) concrete context.

Imagine we want to ask a question: how is the number of guests changing in time?

Our video is going to have a bunch of information in it that’s not relevant to our question (for example, the hair colors of the people at the party). To answer the question, we need to identify what information in the video is *relevant* and discard the rest. That’s moving up a rung on the ladder of abstraction.

Once we identify that relevant information, we can start to talk about representing it in mathematical language, possibly moving up another rung on the ladder.

“But somewhere, not sure where, that level of abstraction would be common place. Say grade 10. Is that wrong?”

I’d say that’s wrong, depending on what you mean. You never see information initially presented that way outside the math classroom. It’s probably still dangerous to start working at that level of abstraction. It’s possible that tenth grades are so skilled at constructing functional abstractions that they can do it trivially, but I sure haven’t met many. Maybe one or two students per year?

## Andy

September 24, 2012 - 8:25 pm -I just presented something similar using a helicopter, ambulance and a “hospital” ( spray bottle as props. The presentation was that the helicopter took off at the rate of 15 feet per second. The students took the helicopter and moved it to compare a table and graph. I asked what would it mean if we used the 15x+10. My students argued if it was possible to start a helicopter at 10 feet in the air, so I presented the issue of the helicopter being on call for police chases. I ended up liking the context of an ambulance on call better. I used this not trying to give my problems context, but to give my students a starting point that I can make more abstract. Perhaps this is psuedocontext, but it seemed to be really helpful.

## Nathan Kraft

September 26, 2012 - 11:26 am -Just saw this article on Education Week. Made me think of this post.

Studies Find Payoff in ‘Personalizing’ Algebra.

http://www.edweek.org/ew/articles/2012/09/26/05personalize_ep.h32.html?tkn=VXCF9%2FGwGEwqob5w42Qm3zUqP9uCwwmsnyD8&cmp=clp-sb-ascd

## Chris Robinson

September 26, 2012 - 4:23 pm -Start with a guess, a gut instinct. Discuss what’s important, what’s not important. Abstract slowly, and talk about why you’re doing what your doing. Climb until you are at that rung where you can apply the pure math. Throughout this abstraction process, the content isn’t really the focus. It’s the problem solving process that is really important here. I’ve come to the conclusion that the math practices of the CCSSM are much more desirable for students to become comfortable with. The content will then come along for the ride.

## joseph

September 27, 2012 - 7:39 am -William wins.

“What if, instead of writing a story [1], you took a picture of a couple of loose tennis balls, and a stack of tennis ball cans. Ask your students to estimate how many tennis balls there are in the photo.”

Make the students do it how they normally do it. Then have them write down the math they just did (or abstracted without realizing it): 3 x #cans + loose ones = total

Make them double-check that they’ve included all units: 3 balls/can x #cans + loose balls= total balls (My physics background urges the importance of this step. The “cans” unit cancels out leaving only “balls.” BTW, I try to do as few ball-related problems as possible with high schoolers.)

They just abstracted it for themselves. Now help them clean up their equations into pretty algebra notation.

## Nico

September 27, 2012 - 5:08 pm -It’s nice to see the article in edweek Nathan. Thanks. I also appreciate that ideas are being shared.

I just want to make mention of something Joseph has said,

‘…the math they just did (or abstracted without realizing it)’

I think that this is the sort of victory we are constantly searching for. They are doing something without even having seen a variable, a constant, coefficient or an equal sign. If I say 87 packages of tennis ball, or they see a picture of 87 packages they will start ‘abstracting without realizing it’.

That’s the victory.

They do the abstraction without me forcing new symbols on them. They might even ask to short form the word packages to a single letter. Another victory.

Ok. This sidetrack was fun. But enough about balls, packages and cans. Who started these comments anyway? This doctoral student is trying to get some work done. More LOA Dan, please…and thank you.